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Full text of "Treatise On Analysis Vol-Ii"

138       XIII    INTEGRATION

Now the sets Apt9tf are closed (3.11.4), hence BM?r is integrable (13.8.7). More-
over, by hypothesis, for each integer r ^ 1 we have ^/f) Bw>r\ = 0, and the

sequence (B^Xg! is decreasing, so that by (13.8.7) we have lim yu(Bn>r) = 0.

n-oo

Hence there exists an integer nr such that n(Bnrtf) g s/2r+2. Let B = (J BMr,r;

r

the set B is integrable, and we have //(B) g f] s/2r+2 = Je by (13.8.7). The

r=l

set C = K0 - B is also integrable, and by definition the sequence (fn \ C)
converges uniformly to f\ C. Hence we may take K' to be any compact
subset of C such that n(C - K') ^ & (13.7.9).

(13.9.11) Let (fn) be any sequence of measurable functions on X with values in
R. Then the functions inffn> sup/M, lim sup/M and lim inffn are measurable.

n                 n                   ./i-* oo                               /i-* oo

For sup/, is the limit of the increasing sequence of functions gn =  sup ft,

n                                                                                     lin

which are measurable (13.9.7); and lim sup/n is the limit of the decreasing

w* oo

sequence of functions sup/n+p, which are measurable, by what has just been

p*o
proved.

(13.9.11.1) The propositions (13.9.6), (13.9.7), (13.9.9), and (13.9.11) remain
valid when the word "measurable" is replaced throughout by "universally
measurable." On the other hand, a function cannot be universally measurable
unless it is everywhere defined, because any nonempty subset N of X has
measure //(N) ^ 0 if for example \JL is the Dirac measure at a point of N.

In particular (12.7.8), every function belonging to/ (resp. ff1) is universally
measurable. If / is any lower semicontinuous function on X, then by
writing

/ = inf (sup (/, - n)}      and       sup (/, - n) = ~ n + sup (/ + n, 0)

n

and using (13.9.6), it follows that

(13L9.11.2)   Every (upper or lower) semicontinuous function on X with values
in R is universally measurable.

The same will be true of all functions obtained from semi-continuous
functions by repeated applications of the operations in (13.9.6), (13.9.7), and
(13.9.11).ictions u \ Ln is continuous.